# The axiom of choice

Content created by Egbert Rijke, Fredrik Bakke and Jonathan Prieto-Cubides.

Created on 2022-02-16.
Last modified on 2024-04-11.

module foundation.axiom-of-choice where

Imports
open import foundation.dependent-pair-types
open import foundation.function-extensionality
open import foundation.functoriality-propositional-truncation
open import foundation.inhabited-types
open import foundation.postcomposition-functions
open import foundation.projective-types
open import foundation.propositional-truncations
open import foundation.sections
open import foundation.split-surjective-maps
open import foundation.surjective-maps
open import foundation.universe-levels

open import foundation-core.equivalences
open import foundation-core.fibers-of-maps
open import foundation-core.function-types
open import foundation-core.functoriality-dependent-pair-types
open import foundation-core.identity-types
open import foundation-core.precomposition-functions
open import foundation-core.sets


## Idea

The axiom of choice asserts that for every family of inhabited types B indexed by a set A, the type of sections of that family (x : A) → B x is inhabited.

## Definition

### The axiom of choice restricted to sets

AC-Set : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)
AC-Set l1 l2 =
(A : Set l1) (B : type-Set A → Set l2) →
((x : type-Set A) → is-inhabited (type-Set (B x))) →
is-inhabited ((x : type-Set A) → type-Set (B x))


### The axiom of choice

AC-0 : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)
AC-0 l1 l2 =
(A : Set l1) (B : type-Set A → UU l2) →
((x : type-Set A) → is-inhabited (B x)) →
is-inhabited ((x : type-Set A) → B x)


## Properties

### Every type is set-projective if and only if the axiom of choice holds

is-set-projective-AC-0 :
{l1 l2 l3 : Level} → AC-0 l2 (l1 ⊔ l2) →
(X : UU l3) → is-set-projective l1 l2 X
is-set-projective-AC-0 ac X A B f h =
map-trunc-Prop
( ( map-Σ
( λ g → ((map-surjection f) ∘ g) ＝ h)
( precomp h A)
( λ s H → htpy-postcomp X H h)) ∘
( section-is-split-surjective (map-surjection f)))
( ac B (fiber (map-surjection f)) (is-surjective-map-surjection f))

AC-0-is-set-projective :
{l1 l2 : Level} →
({l : Level} (X : UU l) → is-set-projective (l1 ⊔ l2) l1 X) →
AC-0 l1 l2
AC-0-is-set-projective H A B K =
map-trunc-Prop
( map-equiv (equiv-Π-section-pr1 {B = B}) ∘ tot (λ g → htpy-eq))
( H ( type-Set A)
( Σ (type-Set A) B)
( A)
( pr1 , (λ a → map-trunc-Prop (map-inv-fiber-pr1 B a) (K a)))
( id))